I'm trying to understand a statement from the book "Perturbation Analysis of Optimization Problems", by Bonnans and Shapiro. Let me start by providing some context. In page 148, the authors write down the optimality conditions for the problem

$$\min_{x\in Q} f(x) \text{ subject to } G(x)\in K,$$

where $Q$ and $K$ are convex subsets of Banach spaces $X$ and $Y$, respectively, and $f\colon X\to \mathbb{R}$, and $G\colon X\to Y$.

In equation 3.8, the authors say that for some feasible point $x_0$, the optimality conditions are

$$x\in\text{argmin}_{x\in Q} L(x,\lambda) \text{ and } \lambda\in N_K(G(x_0)),$$

Where $L(x,\lambda)\colon= f(x) + \langle \lambda, G(X)\rangle$ is the lagrangian associated with the problem, and $N_K(G(x_0))$ is the normal cone of $K$ at the point $x_0$.

After that, in equation 3.9, they say that when $K$ is a cone, the condition $\lambda\in N_K(G(x_0))$ is equivalent to

$$ G(x_0)\in K,\, \lambda \in K^\circ, \text{ and } \langle \lambda, G(x_0)\rangle =0,$$

and that's where I'm struggling. I see that these conditions resemble the classic KKT conditions and they correspond to primal feasibility, dual feasibility, and complementarity, respectively. But I want to figure this out formally and hopefully be able to have some geometric interpretation of it.

Can someone help me out with this?

Thanks in advance